In a village the average age of n people is 42 years. Later it was found that the age of one person had been recorded as 20 years less than the actual age. After correcting this mistake the average age increased by 1 year. What is the value of n?

Introduction:
This question explores how an error in one value affects the average of a group and how to reverse the effect to find the number of people in the group. The key idea is that changing one data point changes the total sum, and therefore changes the average in a predictable way.

Given Data / Assumptions:
- There are n people in a village. - The recorded average age is 42 years. - One person age was recorded as 20 years less than the actual value. - After correcting the mistake, the new average age is 43 years. - We must find the value of n.

Concept / Approach:
The average age is total age divided by the number of people. Initially, using the incorrect data, the total age is taken as 42n. When the error is corrected, the total age increases by 20 years, because the actual age is 20 years more than what was recorded. The new total, divided by n, gives the corrected average of 43. Equating these expressions allows us to determine n.

Step-by-Step Solution:
Step 1: Let the initial (incorrect) total age of all n people be S. Step 2: Given that the average age is 42, we have S = 42n. Step 3: One person age was recorded as 20 years less than the true age, so the true total age should be S + 20. Step 4: After correcting the mistake, the average age becomes 43. Step 5: Therefore, the corrected total age is (correct average) * (number of people) = 43n. Step 6: This corrected total must also equal S + 20. Step 7: Substitute S = 42n into S + 20 = 43n, giving 42n + 20 = 43n. Step 8: Rearrange to get 20 = 43n - 42n = n. Step 9: Therefore, n = 20.

Verification / Alternative Check:
If n = 20, the initial total age is 42 * 20 = 840 years. The corrected total should be larger by 20, so it becomes 860 years. The corrected average is then 860 / 20 = 43 years, which matches the problem statement. This confirms that the value n = 20 fits all the given conditions.

Why Other Options Are Wrong:
Options 21, 22, 24 and 25 do not satisfy the equation 42n + 20 = 43n. Substituting any of these values for n leads to a mismatch between the corrected total and the required average of 43 years.

Common Pitfalls:
Students sometimes incorrectly subtract 20 from the total instead of adding it, or they mix up the old and new averages. It is important to recognize that the incorrect recorded age was too low, so the correction increases the total sum. Keeping track of which total corresponds to which average prevents algebraic mistakes.

Final Answer:
The number of people in the village is 20.